Let us mention that this construction can capture
the extended (logarithmic) flows of the 1D/bigraded
Toda hierartchy [1]}, [2]} as well.
Let \(\tilde{T} = \lbrace \tilde{T}_k\rbrace _{k=1}^\infty \) 
be time variables of the extended flows and deform \(U_0\)  as
\(U_0(\tilde{T})= \exp \left(\sum _{k=1}^\infty \tilde{T}_k\Lambda ^{ka}\log \Lambda \right)U_0\exp \left(- \sum _{k=1}^\infty \tilde{T}_k\Lambda ^{-kb}\log \Lambda \right).\) 
